The self-similar solution of the (1+1)-dimensional optical paraxial diffractive system is found by employing the self-similarity method, leading to the discovery of a beam solution composed of Scorer functions, for which reason we refer to it as the Scorer beam. Through theoretical analysis and numerical simulation, we explore several characteristics of both infinite- and finite-energy Scorer beams, particularly focusing on their non-diffractive and self-accelerating properties. Additionally, we compare the Scorer beam with â€" and distinguish from â€" the related Airy beam. This investigation provides both theoretical and numerical insights that may guide the experimental realization of Scorer beams.